Coordinate transformation from Cartesian to Frenet Frame

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Can anyone tell me how to convert from Cartesian to Frenet Frame for a vehicle driving on a curved road? In other words how do I convert from (x,y) -> (s,d) where is along the curve and d is perpendicular to the curve?
Is there a Matlab code someone can share?
For example if the vehicle is 1 m off centerline and driving along the arc of a circle at speed 1 m/s, then the coordinates at every 1 sec in Cartesian and Frenet frame are given by:
x = cos(theta) ... [with the appropriate scalings for radius and speed]
y = sin(theta) ... [with the appropriate scalings for radius and speed]
s = (0, 1, 2, ...)
d = (1, 1, 1, ...)
I have been looking into Matlab codes but the solutions I got are in the form of (T, N, B) - the tangent, normal and binormal. How do I convert them to distance along the centerline and perpendicular to the centerline?
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M I on 21 May 2021
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Answers (1)

Cameron Stabile
Cameron Stabile on 26 Jan 2023
Hi Suvo,
The referencePathFrenet feature from the Navigation Toolbox might be of use to you. The feature fits a piece-wise clothoid spline between a set of or waypoints, after which you can convert between Cartesian and Frenet space.
There are a number of tools at your disposal, which loosely fall into the following categories:
Projection XY point to Curve:
Conversion between Cartesian and Frenet
Evaluating Curve at Arclength (S)
There are also a number of examples that show how this can be applied to road-based planners, a good one to get you started could be Highway Trajectory Planning Using Frenet Reference Path.
Hope this helps,


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